# Intersection of Symmetric Relations is Symmetric

## Theorem

The intersection of two symmetric relations is also a symmetric relation.

## Proof

Let $\mathcal R_1$ and $\mathcal R_2$ be symmetric relations on a set $S$.

Let $\mathcal R_3 = \mathcal R_1 \cap \mathcal R_2$.

Then:

 $\displaystyle$  $\displaystyle \left({x, y}\right) \in \mathcal R_3$ $\displaystyle$ $\implies$ $\displaystyle \left({x, y}\right) \in \mathcal R_1 \cap \mathcal R_2$ by definition of $\mathcal R_3$ $\displaystyle$ $\implies$ $\displaystyle \left({x, y}\right) \in \mathcal R_1 \land \left({x, y}\right) \in \mathcal R_2$ Definition of Intersection $\displaystyle$ $\implies$ $\displaystyle \left({y, x}\right) \in \mathcal R_1 \land \left({y, x}\right) \in \mathcal R_2$ $\mathcal R_1$ and $\mathcal R_2$ are symmetric $\displaystyle$ $\implies$ $\displaystyle \left({y, x}\right) \in \mathcal R_1 \cap \mathcal R_2$ Definition of Intersection $\displaystyle$ $\implies$ $\displaystyle \left({y, x}\right) \in \mathcal R_3$ by definition of $\mathcal R_3$

$\blacksquare$